Average Error: 28.6 → 28.7
Time: 35.4s
Precision: 64
\[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]
↓
\[\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]
\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}
↓
\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r51221 = x;
        double r51222 = y;
        double r51223 = r51221 * r51222;
        double r51224 = z;
        double r51225 = r51223 + r51224;
        double r51226 = r51225 * r51222;
        double r51227 = 27464.7644705;
        double r51228 = r51226 + r51227;
        double r51229 = r51228 * r51222;
        double r51230 = 230661.510616;
        double r51231 = r51229 + r51230;
        double r51232 = r51231 * r51222;
        double r51233 = t;
        double r51234 = r51232 + r51233;
        double r51235 = a;
        double r51236 = r51222 + r51235;
        double r51237 = r51236 * r51222;
        double r51238 = b;
        double r51239 = r51237 + r51238;
        double r51240 = r51239 * r51222;
        double r51241 = c;
        double r51242 = r51240 + r51241;
        double r51243 = r51242 * r51222;
        double r51244 = i;
        double r51245 = r51243 + r51244;
        double r51246 = r51234 / r51245;
        return r51246;
}
↓
double f(double x, double y, double z, double t, double a, double b, double c, double i) {
        double r51247 = x;
        double r51248 = y;
        double r51249 = z;
        double r51250 = fma(r51247, r51248, r51249);
        double r51251 = 27464.7644705;
        double r51252 = fma(r51250, r51248, r51251);
        double r51253 = 230661.510616;
        double r51254 = fma(r51252, r51248, r51253);
        double r51255 = t;
        double r51256 = fma(r51254, r51248, r51255);
        double r51257 = 1.0;
        double r51258 = a;
        double r51259 = r51248 + r51258;
        double r51260 = b;
        double r51261 = fma(r51259, r51248, r51260);
        double r51262 = c;
        double r51263 = fma(r51261, r51248, r51262);
        double r51264 = i;
        double r51265 = fma(r51263, r51248, r51264);
        double r51266 = r51257 / r51265;
        double r51267 = r51256 * r51266;
        return r51267;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus i

Derivation

  1. Initial program 28.6

    \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644704999984242022037506103515625\right) \cdot y + 230661.5106160000141244381666183471679688\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\]

  2. Simplified28.6

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]
  3. Using strategy rm

  4. Applied div-inv28.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}}\]

  5. Final simplification28.7

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644704999984242022037506103515625\right), y, 230661.5106160000141244381666183471679688\right), y, t\right) \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y + a, y, b\right), y, c\right), y, i\right)}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.764470499998) y) 230661.510616000014) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))